Herman J. Woltring

01-27-1992, 06:13 AM

Dear Biomch-L readers,

In another posting today, reference was made to RoboTech@uscvm.bitnet which,

it did appear, is not a list on robotics. Another query, onto the NA-net

list on Numerical Analysis last week was more succesful. This morning, I

received the latest issue with the response quoted below. Again, it appears

that similar things are (re)discovered in rather different fields, and that,

instead of reinventing the wheel and the gunpowder, it makes sense to look

around in other but potentially related disciplines ...

I have looked up the Bortz and Miller papers which provide some interesting

and complementary ideas. In particular, their interest is to assess attitude

angles from a measured rotation velocity vector; in addition to a concise

angular representation from a given attitude matrix, my interest is just the

opposite: to assess the rotation velocity and acceleration vectors from the

attitude matrix and/or angles.

Herman J. Woltring, Eindhoven/NL

- - - - - - - - - - - - - - - - - - - - - - - -

From: Daniel Johnson

Date: Mon, 20 Jan 1992 08:52:15 -0600

Subject: Re: 3-D Attitude "Vectors"

Sender: NA-net Numerical Analysis Digest, Vol. 92, Nr. 4

Herman J. Woltring asks about references to representing rotations as a

three-dimensional vector where the direction is the axis of rotation, and

the length is the amount of rotation. As a navigation house, we tend to

collect different ways of parametrizing rotations. We refer to his sug-

gestion as the "Bortz Rotation Vector", based on a paper by John Bortz in

1971:

A New Mathematical Formulation for Strapdown Inertial Navigation

John E. Bortz

IEEE Transactions on Aerospace and Electronic Systems

January 1971, Vol. AES-7, No. 1, pp 61-66

He refers to a report by J. Laning in 1949 which I have not looked at:

The vector analysis of finite rotations and angles

J. H. Laning, Jr.

MIT/IL Special Rept. 6398-S-3, 1949

Mass. Inst. of Tech., Cambridge

Some more recent references are

A New Strapdown Attitude Algorithm

Robin B. Miller

J. Guidance, Vol. 6, No. 4, July-Aug 1983, pp 287-291

An Accurate Strapdown Direction Cosine Algorithm

J. W. Jordan

NASA TN D-5384, Sept, 1969

The Bortz formulation is nice in that it has no singularities, has exactly

three free parameters, and has a closed form differential equation for the

rotation angle as a function of angular velocity. By taking lower order

approximations of that equation, it is possible to generate simple

algorithms of acceptable error for updating attitude given the angular

velocity. Typically it is used only for deriving the algorithms, however.

The actual attitude representations used internally in navigation systems

are still either the direction cosine matrix or quaternions.

Dr. Daniel P. Johnson Honeywell Systems and Research Center

e-mail: drdan@src.honeywell.com phone: 612-782-7427

US mail: MN65-2500, 3660 Technology Drive, Minneapolis, MN 55418

------------------------------

In another posting today, reference was made to RoboTech@uscvm.bitnet which,

it did appear, is not a list on robotics. Another query, onto the NA-net

list on Numerical Analysis last week was more succesful. This morning, I

received the latest issue with the response quoted below. Again, it appears

that similar things are (re)discovered in rather different fields, and that,

instead of reinventing the wheel and the gunpowder, it makes sense to look

around in other but potentially related disciplines ...

I have looked up the Bortz and Miller papers which provide some interesting

and complementary ideas. In particular, their interest is to assess attitude

angles from a measured rotation velocity vector; in addition to a concise

angular representation from a given attitude matrix, my interest is just the

opposite: to assess the rotation velocity and acceleration vectors from the

attitude matrix and/or angles.

Herman J. Woltring, Eindhoven/NL

- - - - - - - - - - - - - - - - - - - - - - - -

From: Daniel Johnson

Date: Mon, 20 Jan 1992 08:52:15 -0600

Subject: Re: 3-D Attitude "Vectors"

Sender: NA-net Numerical Analysis Digest, Vol. 92, Nr. 4

Herman J. Woltring asks about references to representing rotations as a

three-dimensional vector where the direction is the axis of rotation, and

the length is the amount of rotation. As a navigation house, we tend to

collect different ways of parametrizing rotations. We refer to his sug-

gestion as the "Bortz Rotation Vector", based on a paper by John Bortz in

1971:

A New Mathematical Formulation for Strapdown Inertial Navigation

John E. Bortz

IEEE Transactions on Aerospace and Electronic Systems

January 1971, Vol. AES-7, No. 1, pp 61-66

He refers to a report by J. Laning in 1949 which I have not looked at:

The vector analysis of finite rotations and angles

J. H. Laning, Jr.

MIT/IL Special Rept. 6398-S-3, 1949

Mass. Inst. of Tech., Cambridge

Some more recent references are

A New Strapdown Attitude Algorithm

Robin B. Miller

J. Guidance, Vol. 6, No. 4, July-Aug 1983, pp 287-291

An Accurate Strapdown Direction Cosine Algorithm

J. W. Jordan

NASA TN D-5384, Sept, 1969

The Bortz formulation is nice in that it has no singularities, has exactly

three free parameters, and has a closed form differential equation for the

rotation angle as a function of angular velocity. By taking lower order

approximations of that equation, it is possible to generate simple

algorithms of acceptable error for updating attitude given the angular

velocity. Typically it is used only for deriving the algorithms, however.

The actual attitude representations used internally in navigation systems

are still either the direction cosine matrix or quaternions.

Dr. Daniel P. Johnson Honeywell Systems and Research Center

e-mail: drdan@src.honeywell.com phone: 612-782-7427

US mail: MN65-2500, 3660 Technology Drive, Minneapolis, MN 55418

------------------------------